Definition and overview of chemometrics

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Definition and overview of chemometrics
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Paul Geladi
Head of Research NIRCE Chairperson NIR Nord Unit
of Biomass Technology and Chemistry Swedish
University of Agricultural Sciences Umeå Technobot
hnia Vasa paul.geladi _at_ btk.slu.se paul.geladi
_at_ syh.fi
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Project geography
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Chemometrics

• Mathematics
• Statistics
• Computer Science
• In Chemistry

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Similar fields

• Biometrics 1900
• Psychometrics 1930
• Econometrics 1950
• Technometrics 1960

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Chemometrics

• Design of Experiments (DOE)
• Exploratory Data Analysis
• Classification
• Regression and Calibration

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Design of Experiments

• Most important where possible
• Uses
• ANOVA
• F-test
• t-test
• Plots
• Response Surfaces

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Design of Experiments

• y b0 b1x1 b2x2 ...bKxK b11x12
• b22x22 ... bKKxK2 b12x1x2 ... e
• Factors x1, x2,...xK changed systematically
• Response y measured and modeled

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Exploratory Data Analysis

• Design not possible
• Sampling situations
• Find structure
• Find groupings
• Find outliers

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Classification

• Check for groupings UNSUPERVISED
• Existing groupings SUPERVISED
• Visualize groupings
• Classify
• Test

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Regression / Calibration

• Two types of variables X / y
• Relationship linear / nonlinear
• Model
• Diagnostics
• Residual

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y
x
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Multivariate Data Analysis
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Multivariate Data Analysis

• Sampled data and design with too many reponses
• Mining
• Hospitals
• Agriculture
• Food industry
• More

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Nomenclature

• Samples are objects
• What is measured on the object is a variable

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34.92
Spectrum
K
1
1
Samples
Vectors
I
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A vector is a collection of numbers. It is
always a column vector.
12 3.6 11.1 5.9 34 0.5 1.4 17
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12 3.6 11.1 5.9 34 0.5 1.4 17
The transpose of a vector is a row
vector. Symbols for transpose are and T. a or
aT.
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Particle size, 1 sample
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Small particles, 35 samples
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The Data Matrix
K
A data matrix is a vector of vectors
I
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Size histograms, all samples
Particle area
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Times in batch reaction
NIR wavelengths
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Geometry of multivariate space
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Problem

• I and K can be large
• Correlation
• Univariate statistics does not apply

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3 variables blood oxygen, iron, hemoglobin
I patients
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Hb
Fe
O2
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Hb
Fe
O2
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Hb
Fe
O2
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Hb
Fe
O2
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Hb
Fe
O2
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Hb
Fe
O2
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Hb
Fe
O2
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Hb
Fe
O2
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Hb
Fe
O2
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Properties of multivariate space

• Rotation
• vectors unchanged / distance unchanged
• Translation
• vectors changed / distance unchanged
• Rescaling / change units
• all changes

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Consequences

• We can move the coordinate sytem around
• The relative distances between objects do not
  change
• We can rotate the coordinate system
• Scale changes are important
• Move coordinate system to center of data
• Scale properly

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Vectors (physics)
x x1, x2, x3
x ( x12 x22 x32 ) 1/2
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Geometry
c2 a2 b2
c
a
b
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Vectors (K dimensions)
x x1, x2,..., xK
x ( x12 x22 ... xK2 ) 1/2
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Problem

• We can not see in more than 3 dimensions
• Paper, computer screen 2-2.5 dimensions

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Hb
Fe
O2
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Hb
Fe
O2
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Projection

• 2D plane (screen, paper)
• Many projections possible
• Find a good one
• Find a few good ones
• What is good?